Where θ and φ are the Greek characters theta and phi respectively.
Six basic trig functions (SOHCAHTOA)
tan θ = opposite / adjacent
cot θ = adjacent / opposite
sin θ = opposite / hypotenuse
csc θ = hypotenuse / opposite
cos θ = adjacent / hypotenuse
sec θ = hypotenuse / adjacent
identites (some of these are useful since a calculator does not have csc(), sec() or cot())
tan θ = 1 / cot θ
sin θ = 1 / csc θ
cos θ = 1 / sec θ
sin2θ + cos2θ = 1
cot θ = 1 / tan θ
csc θ = 1 / sin θ
sec θ = 1 / cos θ
Inverse functions ( go from ratios to angles) arctangent, arcsine, arccosine
atan (opp / adj ) = θ sometimes written as tan-1 (opp / adj ) = θ
asin ( opp / hyp ) = θ sometimes written as sin-1 ( opp / hyp ) = θ
acos ( adj/ hyp ) = θ sometimes written as cos-1 ( adj / hyp ) = θ
Given θ in degrees, here are the functions of the complementary angle in degrees
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tan θ = cot (90 - θ)
sin θ = cos (90 - θ )
sec θ = csc ( 90 - θ )
cot θ = tan ( 90 - θ )
cos θ = sin ( 90 - θ )
csc θ = sec ( 90 - θ )
Given θ in radians, here are the functions of the complementary angle in radians
tan θ = cot (π/2 - θ)
sin θ = cos (π/2 - θ )
sec θ = csc (π/2 - θ )
cot θ = tan (π/2 - θ )
cos θ = sin (π/2 - θ )
csc θ = sec ( π/2 - θ )
Symmetry
- sin ( θ ) = sin ( - θ )
- tan ( θ ) = tan ( - θ )
cos ( θ ) = cos ( - θ )
-csc ( θ ) = csc ( - θ )
-cot ( θ ) = cot ( - θ )
sec ( θ ) = sec ( - θ )
--------------On the graph-----------
functions of the quadrantal angles (i.e. angles ON the x and y axes)
tan 00 = 0
tan 900 = undefined
tan 1800 = 0
tan 2700 = undefined
tan 3600 = 0
sin 00 = 0
sin 900 = 1
sin 1800 = 0
sin 2700 = -1
sin 3600 = 0
cos 00 = 1
cos 900 = 0
cos 1800 = -1
cos 2700 = 0
cos 3600 = 1
Signs of trig functions in quadrants 1,2,3 and 4 All Students Take Calculus
Quadrant 1
All trig functions are + in first Quadrant
Quadrant 2
ONLY sin θ = + and csc θ = + in second quadrant, all others are negative
Quadrant 3
Only tan θ = + and cot θ = + in third quadrant, all others are negative
Quadrant 4
Only cos θ = + and sec θ = + in fourth quadrant, all others are negative
--------------------------------------------------Functions of special angles 30 degrees, 45 degrees and 60 degrees
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45 degrees ( or π/4 radians)
tan 45° = 1
sin 45° = 1 / √2
60 degrees (or π/3 radians )
tan 60° = √3sin 60° = √3 / 2
cos 45° = 1 / √2
cos 60° = 1 / 2
30 degrees ( or π / 6 radians)
tan 30° = √3 / 3
sin 30° = 1 / 2
Converting between degrees and radians.
( θ° ) ( π rads / 180° ) = radian angle
( θ rads ) ( 180° / π rads ) = angle in degrees
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cos 30° = √3 / 2
Arc Length on a circle = the angle subtended in radians X the radius
or
S= θ R
Linear velocity of a point on the edge of a spinning circle = angular velocity X radius
or LV = AV X R
To calculate the reference angle θ1 given the actual angle θ0 on a graph
Conversion of degrees minutes seconds between degrees in decimal
Given 100° 45' 34"
degrees in decimal = 100 + 45 / 60 + 34 / 3600
Given 152. 5639°
step 1. multiply .5639 X 60 The whole part of the answer is the number of minutes
step 2. then multiply the decimal part of that result by 60. The answer (rounded to a
4
whole number) is the seconds part.
π
θ
φ
√
°
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